If the number \(k\)  remains fixed, then we no longer have such a sharp threshold as above. The following was also proved in [1]}. If \(\frac{d}{N} \rightarrow \delta \quad \mbox{as }d\rightarrow \infty \)  with a number \(\delta \in [0,1]\)  and if \(k\in {\mathbb {N}}\)  is fixed, then
\(\lim _{d\rightarrow \infty } \frac{{\mathbb {E}}\,f_k(C_N)}{\binom{N}{k}} = \left\lbrace \begin{array}{ll} 1 & \mbox{if } 1/2<\delta \le 1,\\[1mm](2\delta )^k & \mbox{if } 0\le \delta < 1/2.\end{array}\right.\) 
